Wavefronts and parallels in Euclidean space

J.W. Bruce

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

Given a smooth plane curve or surface in 3 its parallels consist of those curves or surfaces a fixed distance d down the normals in a fixed direction. Generically they have Legendre singularities. We are concerned here with the way in which these parallels change as we alter the distance d. (Alternatively the manner in which wave-fronts change as they evolve from an initial smooth wavefront.) This problem was considered in (1) by V. I. Arnold. In a very beautiful paper he describes the generic evolution of wavefronts but does not prove that for a generic initial wavefront in 2 or 3 the evolution is of the type described there. This we do here, using the tool of transversality. A more positive outcome of our investigation is that some of Arnold's generic forms do not occur (those corresponding to A2 singularities).
Original languageEnglish
Pages (from-to)323-333
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume93
Issue number2
DOIs
Publication statusPublished - 1983

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Wave Front
Euclidean space
Singularity
Transversality
Plane Curve
Legendre
Curve

Cite this

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Wavefronts and parallels in Euclidean space. / Bruce, J.W.

In: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 93, No. 2, 1983, p. 323-333.

Research output: Contribution to journalArticle

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